Matrix Transpose Calculator
Calculate the transpose of a matrix with step-by-step explanations and matrix properties
Matrix Input
Original Matrix A
Enter matrix elements. Use decimal numbers (e.g., 0.5, -2.3)
Quick Examples
Matrix Transpose Results
Matrix Properties
- Symmetric (A = A^T):No
- Skew-Symmetric (A = -A^T):No
- Diagonal Matrix:No
- Same Shape After Transpose:Yes
Transpose Properties
Definition: (A^T)[j][i] = A[i][j]
Size Change: m×n → n×m
Double Transpose: (A^T)^T = A
Sum Property: (A + B)^T = A^T + B^T
Step-by-Step Solution
Matrix Transpose A^T
This is the transpose matrix A^T where rows and columns are interchanged
Visual Transformation
Original Matrix A
3×3 matrix
Transpose
Transpose A^T
3×3 matrix
How to Transpose
Flip Rows & Columns
Row i becomes column i
Apply Formula
A^T[j][i] = A[i][j]
Check Properties
Verify symmetry if square
Key Properties
(A^T)^T = A: Double transpose returns original
(A + B)^T = A^T + B^T: Distributive over addition
(AB)^T = B^T A^T: Product rule (reverse order)
det(A^T) = det(A): Determinant unchanged
Understanding Matrix Transpose
What is Matrix Transpose?
The transpose of a matrix A, denoted as A^T, is formed by interchanging the rows and columns of the original matrix. If A is an m×n matrix, then A^T is an n×m matrix where the element at position (i,j) in A becomes the element at position (j,i) in A^T.
How to Calculate Transpose
- •Take each row of the original matrix
- •Make it a column in the transpose matrix
- •Apply the formula: A^T[j][i] = A[i][j]
Special Matrix Types
- Symmetric Matrix: A = A^T
- Skew-Symmetric: A = -A^T
- Orthogonal Matrix: A^T A = I
- Normal Matrix: A A^T = A^T A
Applications
- •Solving systems of linear equations
- •Data analysis and machine learning
- •Computer graphics transformations